On universal quadratic identities for minors of quantum matrices

TL;DR

This paper provides a complete combinatorial characterization of universal quadratic relations among minors in quantum matrices, using path-based models in planar graphs, advancing understanding of quantum matrix algebra.

Contribution

It introduces a new combinatorial framework for identifying universal quadratic identities among minors in quantum matrices, generalizing previous graph models.

Findings

Complete characterization of quadratic relations for quantum minors

Path-based method for verifying identities

Extension to generalized planar graph models

Abstract

We give a complete combinatorial characterization of homogeneous quadratic relations of "universal character" valid for minors of quantum matrices (more precisely, for minors in the quantized coordinate ring $O_q(M_{m,n}(K))$ of $m\times n$ matrices over a field $K$, where $q\in K^\ast$). This is obtained as a consequence of a study of quantized minors of matrices generated by paths in certain planar graphs, called SE-graphs, generalizing the ones associated with Cauchon diagrams. Our efficient method of verifying universal quadratic identities for minors of quantum matrices is illustrated with many appealing examples.